Seeing Structure in Expressions - Factoring GCF's and Quadratics

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SWBAT factor a greatest common factor from an expression and factor quadratic trinomials by grouping.

Big Idea

The same polynomial expression can be presented in several ways; "factoring" an expression means to write it as a product.


20 minutes

Students will come to class with their GCF BINGO cards, which they completed for homework.  In this assignment, they factored a GCF out of 20 expressions and then filled in their bingo card with these GCF's.  After I circulate around the room and check this assignment with the homework rubric, we are ready to play BINGO.  

Using GCF Bingo Selector, I scroll through the slides and call out the GCF's one at a time.  I use a copy of the answer key in a sheet protector and mark which ones I have called with a dry-erase marker so that I can verify who gets bingo.  In my experience, all classroom games go a lot more smoothly when the teacher is willing to be very strict with the rules of the game.  In bingo, this means having students call out each of the GCF's in their win to verify that they have indeed been called.  This has the added benefit of students practicing how to read expressions with exponents.

I typically do three rounds of BINGO and do not have students clear off their cards between rounds.  The winner of each round gets a small prize like a piece of candy.

Direct Instruction - Factoring

20 minutes

Although students were introduced to factoring in Algebra 1, many students will feel as if they never learned it.  Fluency in factoring is essential to understanding the polynomial theorems we explore in this unit so I take time here to make sure that students develop this fluency.

Most textbooks teach quadratic factoring using several strategies (one method for when the lead coefficient is equal to one, another for when it is not equal to one, a third for differences of two squares, etc).  I believe that this leads to much confusion and prefer to focus on a single strategy that works for every factorable quadratic. No matter what form a quadratic expression takes, if it is factorable "factoring by grouping" can get them there.  See Notes on Factoring by Grouping for details on this method.  

Later, when students have factored many quadratic expressions, they will become more adept at recognizing the structure of an expression which will lead to shortcut methods [MP7].

I ask students to take explicit notes on this method and we work through about 5  examples together.  These examples include a wide variety of quadratic expressions, including "special" factoring patterns like the difference of two squares and perfect square trinomials as well as quadratics that have a GCF that should be factored out first. (I refer to the "BINGO factoring" if they forget).  I have included WS Factoring Quadratics as a source of examples that can be worked out on the board.  


Practice and Reinforcement

40 minutes

For practice, I offer students leveled worksheets.   I print 3 different sets of quadratic expressions to factor on 3 different colors of paper.  I explain to students that the green practice problems are entry-level work for students who feel they need to work up to this concept, yellow is about the level I will put on assessments and red is for students who are ready for a challenge.   I instruct students to take the level they are ready for and find someone who is working on the same level to sit next to.  Students may work on the white boards in the classroom, in their notebooks, or on small white boards at their desks according to their preference.

Generally, students respond well to being allowed to choose the level of difficulty they are ready for and most tend to make good choices.  The most important aspect of this practice is that it helps students understand that factoring is just rewriting an expression as a multiplication problem and that they can use a single strategy for every quadratic expression, although many different methods exist [MP7].

As students work through the practice problems, they use the 3-Cup System to let me know when they need my help [MP1]. If a particular group needs a lot of support I suggest that they may have selected the wrong level of practice.  

Exit Ticket and Assignment

10 minutes

After the day's lesson I do a quick check of student's factoring skills with Exit Ticket Factoring Quadratics.  For homework, I assign WS Factoring Quadratic Expressions which is a collection of 20 quadratic factoring problems.  There are a wide variety of expressions on this worksheet and I instruct students to select 12 problems at their level to complete for homework.  The solutions to this assignment will be available on Edmodo.